A filtration on rings of representations of non-Archimedean $G{L}_n$

Abstract

Let $F$ be a $p$-adic field. Let $\mathcal{R}$ be the Grothendieck ring of complex smooth finite-length representations of the groups $\{G{L}_n(F){\}}_{n=0}^{\infty }$ taken together, with multiplication defined in the sense of parabolic induction. We introduce a width invariant for elements of $\mathcal{R}$ and show that it gives an increasing filtration on the ring. Irreducible representations of width $1$ are precisely those known as ladder representations. We thus obtain a necessary condition on irreducible factors of a product of two ladder representations. For such a product we further establish a multiplicity-one phenomenon, which was previously observed in special cases.

11 pages